Ready-to-blossom bases and the existence of geometrically continuous piecewise Chebyshevian B-splines

Marie-Laurence Mazure

doi:10.1016/j.crma.2009.04.026

Numerical Analysis

Ready-to-blossom bases and the existence of geometrically continuous piecewise Chebyshevian B-splines
[B-splines de Chebyshev et bases sur mesure pour les floraisons]

Présenté par : Philippe G. Ciarlet

Marie-Laurence Mazure ¹

¹ Laboratoire Jean-Kuntzmann, Université Joseph-Fourier, BP 53, 38041 Grenoble cedex 9, France

Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 829-834.

Résumé
Abstract

Dans tout espace de splines à sections dans différents espaces de Chebyshev généralisés et matrices de connexion, nous caractérisons l'existence de floraisons (cruciale pour le design) par celle de bases sur mesure définies en termes de zéros. Ce résultat nous permet d'obtenir l'équivalence entre l'existence de floraisons et celle de bases de B-splines sous des hypothèses de différentiabilité aussi faibles que possible.

Existence of blossoms is crucial for design. In a single space, we recently characterised it in terms of ready-to-blossom bases. Such bases are magic, for their use makes existence of blossoms visible at first sight. A similar characterisation is given here for geometrically continuous piecewise Chebyshevian splines (sections in different Extended Chebyshev spaces, connection matrices at the knots). This enables us to re-prove the equivalence between existence of blossoms and existence of B-spline bases under the least possible differentiability assumptions. The existing proof of the latter result was totally different and it strongly relied on the fact that all spline sections were supposed to be $C^{\infty}$ .

Reçu le : 2009-01-11
Accepté le : 2009-04-16
Publié le : 2009-05-23

DOI : 10.1016/j.crma.2009.04.026

Affiliations des auteurs :

Marie-Laurence Mazure ¹

¹ Laboratoire Jean-Kuntzmann, Université Joseph-Fourier, BP 53, 38041 Grenoble cedex 9, France

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     author = {Marie-Laurence Mazure},
     title = {Ready-to-blossom bases and the existence of geometrically continuous piecewise {Chebyshevian} {B-splines}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {829--834},
     publisher = {Elsevier},
     volume = {347},
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     year = {2009},
     doi = {10.1016/j.crma.2009.04.026},
     language = {en},
}

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Marie-Laurence Mazure. Ready-to-blossom bases and the existence of geometrically continuous piecewise Chebyshevian B-splines. Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 829-834. doi : 10.1016/j.crma.2009.04.026. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.04.026/

Bibliographie
Cité par

[1] M.-L. Mazure On the equivalence between existence of B-spline bases and existence of blossoms, Constructive Approximation, Volume 20 (2004), pp. 603-624

[2] M.-L. Mazure Ready-to-blossom bases in Chebyshev spaces (K. Jetter; M. Buhmann; W. Haussmann; R. Schaback; J. Stoeckler, eds.), Topics in Multivariate Approximation and Interpolation, Elsevier, 2006, pp. 109-148

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